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All Formulas At a Glance

🎯 Projectile Motion

vₓ = v₀cosθ (constant)

vy = v₀sinθ − gt

x = v₀cosθ · t

y = v₀sinθ·t − ½gt²

T = 2v₀sinθ/g

H = v₀²sin²θ/2g

R = v₀²sin2θ/g

R_max = v₀²/g (at θ=45°)

📐 Trajectory & Special Cases

y = x tanθ − gx²/2v₀²cos²θ

y = x tanθ(1 − x/R)

R = 4H cotθ

θ & (90°−θ): same R, diff H

H₁/H₂ = tan²θ₁/tan²θ₂

T₁/T₂ = sinθ₁/sinθ₂

R₁/R₂ = sin2θ₁/sin2θ₂

↕ Horizontal Projectile (from h)

T = √(2h/g)

R = u·√(2h/g)

vy = √(2gh) at landing

v_f = √(u² + 2gh)

angle at landing: tanφ = vy/vₓ

🔄 Uniform Circular Motion

v = ωr = 2πr/T = 2πrf

ω = 2πf = 2π/T = v/r

aᶜ = v²/r = ω²r = vω

F = mv²/r = mω²r

T = 2π/ω = 2πr/v

|Δv⃗| = 2v·sin(α/2)

⭕ Vertical Circle

v_top,min = √(gR)

v_bottom,min = √(5gR)

v_b² = v_t² + 4gR

T_b − T_t = 6mg

T at angle θ: T = mv²/R ± mgcosθ

↗ Relative Motion

v⃗_AB = v⃗_A − v⃗_B

|v_AB| = √(vA²+vB²−2vAvBcosα)

Boat (min time): t = d/v_b

Boat (zero drift): sinθ = v_r/v_b

Two projectiles: a_rel = 0

⛰ Inclined Plane Projectile

T = 2v₀sinβ/(gcosα)

R = 2v₀²sinβcos(α+β)/(gcosα)

Max R angle: β = 45° − α/2

R_max = v₀²/[g(1+sinα)]

⚙ Non-Uniform Circular Motion

aᶜ = v²/r (centripetal)

aₜ = dv/dt (tangential)

a_net = √(aᶜ² + aₜ²)

tanφ = aₜ/aᶜ (angle with radius)

Memory Tricks

Remember These Instantly

🟡 "45 is Max"

Range is maximum at 45°. Remember: 45 is the perfect balance between horizontal and vertical. Less angle → too flat. More → too vertical.

🔢 T, H, R ratios for 30-45-60

θ=30°: T=v₀/g, H=v₀²/8g, R=v₀²√3/2g
θ=45°: T=v₀√2/g, H=v₀²/4g, R=v₀²/g
θ=60°: T=v₀√3/g, H=3v₀²/8g, R=v₀²√3/2g

🔵 Complementary = Same R

30° & 60°, 20° & 70°, 15° & 75° — all same range. Different height, different time, SAME range. Simple: sin2θ = sin(180°−2θ).

🔴 Vertical Circle Magic Number

T_bottom − T_top = 6mg. Always. No exceptions. This means if T_top = mg, then T_bottom = 7mg.

⚡ UCM Energy Constant

KE = ½mv² = constant. Work done by centripetal force = 0 (force ⊥ displacement). Speed never changes in UCM.

🌊 Horizontal always constant

In ANY projectile problem (no air resistance): horizontal velocity = constant throughout flight. This is the #1 principle. Never forget.

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1 / 12 (Click card to flip)

What is the time of flight formula for a projectile launched at angle θ?

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T = 2v₀ sinθ / g

Half of T is the time to reach max height

At the highest point of a projectile, what is the vertical velocity?

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vy = 0

Only horizontal component vₓ = v₀cosθ remains

What angle gives maximum horizontal range?

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θ = 45°

R_max = v₀²/g at θ = 45°

For a horizontal projectile from height h, what is the time to hit ground?

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T = √(2h/g)

Independent of horizontal velocity!

Formula for centripetal acceleration?

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aᶜ = v²/r = ω²r

Always directed toward center

What is the relationship between T_bottom and T_top in vertical circle?

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T_b − T_t = 6mg

This is always true — independent of speed

Formula: minimum speed at top of vertical circle?

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v_top,min = √(gR)

So v_bottom,min = √(5gR)

For two simultaneously launched projectiles — what is the relative acceleration?

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a_relative = 0

Both have same g. So from one projectile's frame, other moves in straight line.

Trajectory equation of a projectile?

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y = x tanθ − gx²/(2v₀²cos²θ)

Also: y = x tanθ(1 − x/R)

Which angle gives same range? (Not 45°)

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θ and (90° − θ)

Example: 30° and 60° both give R = v₀²√3/(2g)

Direction of velocity in UCM?

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Always tangent to the circular path

Perpendicular to radius. So v⃗ · r⃗ = 0

Formula for change in velocity vector |Δv⃗| in circular arc of angle α?

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|Δv⃗| = 2v sin(α/2)

Quarter circle: v√2 | Half circle: 2v | Full circle: 0

One-Page Summary

What You Must Know Cold

📌 Core Principles

Horizontal and vertical motions are independent
Horizontal velocity = constant (no air resistance)
Vertical: uniformly accelerated downward (g)
Trajectory is a parabola
In UCM: speed constant, velocity changes
Centripetal acc is always toward center

📌 Number Facts

θ = 45° → max range = v₀²/g
θ = 90° → max height = v₀²/2g
T_b − T_t = 6mg (vertical circle)
v_top,min = √(gR)
v_bot,min = √(5gR)
Complementary angles → same R

📌 Common Mistakes to Avoid

Centripetal = net force, not extra force
Time of flight = independent of horizontal v
Average velocity ≠ average speed in 2D
Acceleration in UCM ≠ 0 (it's centripetal)
Discard negative time root
Sign convention must be consistent

📌 JEE Advanced Must-Knows

Inclined projectile: rotate axes by α
Relative projectile: a_rel = 0 → straight line
Variable acc: use calculus (∫ a dt)
Vertical circle string vs track
Conical pendulum: T = 2π√(Lcosθ/g)

✅ Chapter Complete!

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